On semilinear Cauchy problems with non-dense domain (Q1032712)

The authors consider the Cauchy problem of the following semilinear evolution equation in a Banach space \(X\) \[ u'(t)=A u(t)+F(t,\,u(t)),\quad t\geq 0,\quad u(0)=u_0\in \overline{D(A)}, \] where \(A:D(A)\subset X\to X\) is a closed linear operator whose domain \(D(A)\) is not dense in \(X\), and \(F:[0,\,+\infty)\times \overline{D(A)}\to X\) is a continuous mapping. Using integrated semigroup theory, they research the positivity of the solutions, the Lipschitz perturbation, differentiability of the solutions with respect to the state variable, time differentiability of the the solutions, and the stability of equilibria. Two application examples to partial differential equations are given.